Introduction to operations research 10th edition fred hillier test bank

Test Bank for Chapter Problem 3-1: The Weigelt Corporation has three branch plants with excess production capacity Fortunately, the corporation has a new product ready to begin production, and all three plants have this capability, so some of the excess capacity can be used in this way This product can be made in three sizes large, medium, and small that yield a net unit profit of $420, $360, and $300, respectively Plants 1, 2, and have the excess capacity to produce 750, 900, and 450 units per day of this product, respectively, regardless of the size or combination of sizes involved The amount of available in-process storage space also imposes a limitation on the production rates of the new product Plants 1, 2, and have 13,000, 12,000, and 5,000 square feet, respectively, of in-process storage space available for a day's production of this product Each unit of the large, medium, and small sizes produced per day requires 20, 15, and 12 square feet, respectively Sales forecasts indicate that if available, 900, 1,200, and 750 units of the large, medium, and small sizes, respectively, would be sold per day At each plant, some employees will need to be laid off unless most of the plant’s excess production capacity can be used to produce the new product To avoid layoffs if possible, management has decided that the plants should use the same percentage of their excess capacity to produce the new product Management wishes to know how much of each of the sizes should be produced by each of the plants to maximize profit Formulate a linear programming model for this problem Solution for Problem 3.1: The decision variables can be denoted and defined as follows: xP1L xP1M xP1S xP2L xP2M xP2S xP3L xP3M xP3S = = = = = = = = = number of large units produced per day at Plant 1, number of medium units produced per day at Plant 1, number of small units produced per day at Plant 1, number of large units produced per day at Plant 2, number of medium units produced per day at Plant 2, number of small units produced per day at Plant 2, number of large units produced per day at Plant 3, number of medium units produced per day at Plant 3, number of small units produced per day at Plant Also letting P (or Z) denote the total net profit per day, the linear programming model for this problem is Maximize P = 420 xP1L + 360 xP1M + 300 xP1S + 420 xP2L + 360 xP2M + 300 xP2S + 420 xP3L + 360 xP3M + 300 xP3S, subject to xP1L + xP1M + xP1S  750 xP2L + xP2M + xP2S  900 xP3L + xP3M + xP3S  450 20 xP1L + 15 xP1M + 12 xP1S  13000 20 xP2L + 15 xP2M + 12 xP2S  12000 20 xP3L + 15 xP3M + 12 xP3S  5000 xP1L + xP2L + xP3L  900 xP1M + xP2M + xP3M  1200 xP1S + xP2S + xP3S  750 1 ( xP1L + xP1M + xP1S ) ( xP2L + xP2M + xP2S ) = 750 900 1 ( xP1L + xP1M + xP1S ) ( xP3L + xP3M + xP3S ) = 750 450 and xP1L  0, xP1M  0, xP1S  0, xP2L  0, xP2M  0, xP2S  0, xP3L  0, xP3M  0, xP3S  The above set of equality constraints also can include the following constraint: ( x P2 L + x P2 M + x P2S ) - (x P3 L + xP3M + x P3S ) = 900 450 However, any one of the three equality constraints is redundant, so any one (say, this one) can be deleted Problem 3-2: Comfortable Hands is a company which features a product line of winter gloves for the entire family — men, women, and children They are trying to decide what mix of these three types of gloves to produce Comfortable Hands’ manufacturing labor force is unionized Each full-time employee works a 40-hour week In addition, by union contract, the number of full-time employees can never drop below 20 Nonunion, part-time workers can also be hired with the following union-imposed restrictions: (1) each part-time worker works 20 hours per week, and (2) there must be at least full-time employees for each part-time employee All three types of gloves are made out of the same 100% genuine cowhide leather Comfortable Hands has a long term contract with a supplier of the leather, and receives a 5,000 square feet shipment of the material each week The material requirements and labor requirements, along with the gross profit per glove sold (not considering labor costs) is given in the following table Material Required Labor Required Gross Profit Glove (square feet) (minutes) (per pair) Men’s 30 $8 Women’s 1.5 45 $10 Children’s 40 $6 Each full-time employee earns $13 per hour, while each part-time employee earns $10 per hour Management wishes to know what mix of each of the three types of gloves to produce per week, as well as how many full-time and how many part-time workers to employ They would like to maximize their net profit — their gross profit from sales minus their labor costs Formulate a linear programming model for this problem Solution for Problem 3-2: The decision variables can be denoted and defined as follows: M = number of men’s gloves to produce per week, W = number of women’s gloves to produce per week, C = number of children’s gloves to produce per week, F = number of full-time workers to employ, PT = number of part-time workers to employ (Alternative notation for the decision variables is xM, xW, xC, xF, and xPT, respectively.) Also letting P (or Z) denote the total net profit per week, the linear programming model for this problem is Maximize P = M + 10 W + C – 13(40)F – 10(20) PT, subject to M + 1.5 W + C  5000 30 M + 45 W + 40 C  40(60) F + 20(60) PT F  20 F  PT and M  0, W  0, C  0, F  0, PT  Problem 3-3: Slim-Down Manufacturing makes a line of nutritionally complete, weight-reduction beverages One of their products is a strawberry shake which is designed to be a complete meal The strawberry shake consists of several ingredients Some information about each of these ingredients is given below Calories Total Vitamin from fat Calories Content Thickeners Cost (per tbsp) (per tbsp) (mg/tbsp) (mg/tbsp) (¢/tbsp) 50 20 10 75 100 8 Vitamin supplement 0 50 25 Artificial sweetener 120 15 30 80 25 Ingredient Strawberry flavoring Cream Thickening agent The nutritional requirements are as follows The beverage must total between 380 and 420 calories (inclusive) No more than 20% of the total calories should come from fat There must be at least 50 milligrams (mg) of vitamin content For taste reasons, there must be at least two tablespoons (tbsp) of strawberry flavoring for each tbsp of artificial sweetener Finally, to maintain proper thickness, there must be exactly 15 mg of thickeners in the beverage Management would like to select the quantity of each ingredient for the beverage which would minimize cost while meeting the above requirements Formulate a linear programming model for this problem Solution for Problem 3-3: The decision variables can be denoted and defined as follows: S = Tablespoons of strawberry flavoring, CR = Tablespoons of cream, V = Tablespoons of vitamin supplement, A = Tablespoons of artificial sweetener, T = Tablespoons of thickening agent (Alternative notation for the decision variables is xS, xC, xV, xA, and xT, respectively.) Also letting C (or Z) denote cost, the linear programming model for this problem is Minimize C = 10 S + CR + 25 V + 15 A + T, subject to 50 S + 100 CR + 120 A + 80 T  380 50 S + 100 CR + 120 A + 80 T  420 S + 75 CR + 30 T  0.2 (50 S + 100 CR + 120 A + 80 T) 20 S + 50 V + T ≥ 50 S  2A S + CR + V + A + 25 T = 15 and S  0, CR  0, V  0, A  0, T  Problem 3-4: Back Savers is a company that produces backpacks primarily for students They are considering offering some combination of two different models—the Collegiate and the Mini Both are made out of the same rip-resistant nylon fabric Back Savers has a longterm contract with a supplier of the nylon and receives a 5000 square-foot shipment of the material each week Each Collegiate requires square feet while each Mini requires square feet The sales forecasts indicate that at most 1000 Collegiates and 1200 Minis can be sold per week Each Collegiate requires 45 minutes of labor to produce and generates a unit profit of $32 Each Mini requires 40 minutes of labor and generates a unit profit of $24 Back Savers has 35 laborers that each provides 40 hours of labor per week Management wishes to know what quantity of each type of backpack to produce per week (a) Formulate and solve a linear programming model for this problem on a spreadsheet (b) Formulate this same model algebraically (c) Use the graphical method by hand to solve this model Solution for Problem 3-4: (a) To build a spreadsheet model for this problem, start by entering the data The data for this problem are the unit profit of each type of backpack, the resource requirements (square feet of nylon and labor hours required), the availability of each resource, 5400 square feet of nylon and (35 laborers)(40 hours/laborer) = 1400 labor hours, and the sales forecast for each type of backpack (1000 Collegiates and 1200 Minis) In order to keep the units consistent in row (hours), the labor required for each backpack (in cells C8 and D8) are converted from minutes to hours (0.75 hours = 45 minutes, 0.667 hours = 40 minutes) The range names UnitProfit (C4:D4), Available (G7:G8), and SalesForecast (C13:D13) are added for these data The decision to be made in this problem is how many of each type of backpack to make Therefore, we add two changing cells with range name UnitsProduced (C11:D11) The values in CallsPlaced will eventually be determined by the Solver For now, arbitrary values of 10 and 10 are entered The goal is to produce backpacks so as to achieve the highest total profit Thus, the objective cell should calculate the total profit, where the objective will be to maximize this objective cell In this case, the total profit will be Total Profit = ($32)(# of Collegiates) + ($24)(# of Minis) or Total Cost = SUMPRODUCT(UnitProfit, UnitsProduced) This formula is entered into cell G11 and given a range name of TotalProfit With 10 Collegiates and 10 Minis produced, the total profit would be ($32)(10) + ($24)(10) = $560 The first set of constraints in this problem involve the limited available resources (nylon and labor hours) Given the number of units produced (UnitsProduced in C11:D11), we calculate the total resources required For nylon, this will be =SUMPRODUCT(C7:D7, UnitsProduced) in cell E7 By using a range name or an absolute reference for the units produced, this formula can be copied into cell E8 to calculate the labor hours required The total resources used (TotalResources in E7:E8) must be